Question

With reference to the formal proof of validity of the following argument, state the correct order of rules employed.
1. \( P \cdot Q \)
2. \( ( P \vee R ) \supset S / \therefore P \cdot S \)
3. P
4. \( P \vee R \)
5. S
6. \( P \cdot S \)
A. Conjunction
B. Modus Ponens
C. Addition
D. Simplification
Choose the correct answer from the options given below:

• A. A, B, C, D
• B. B, A, D, C
• C. D, C, B, A
• D. C, B, A, D

Hint:

In formal proofs, remember to apply Simplification to break down conjunctions, Addition to introduce disjunctions, Modus Ponens to apply conditionals, and Conjunction to combine statements.

Answer 1

The Correct Option is C

Step 1: Deconstructing the Formal Proof Structure.
The proof progresses through a sequence of logical operations to derive the conclusion \( P \cdot S \) from the established premises. The sequence is as follows: - D. Simplification: Initial step involves reducing the conjunction \( P \cdot Q \) to its constituent \( P \), utilizing the Simplification rule. - C. Addition: Subsequently, \( P \) is incorporated into \( P \vee R \), adhering to the Addition rule (as \( P \) logically entails \( P \vee R \)). - B. Modus Ponens: The next phase employs Modus Ponens. Given the premise \( (P \vee R) \supset S \) and the derived \( P \vee R \), the result \( S \) is deduced. - A. Conjunction: The final operation combines \( P \) and \( S \) to yield the conclusion \( P \cdot S \), applying the Conjunction rule.
Step 2: Verified Sequence. The logically sound order of operations is confirmed as D, C, B, A.
Final Answer: \[ \boxed{\text{The correct sequence of rules is D, C, B, A.}} \]